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Applying Rational Numbers Chapter Focus You will add, subtract, multiply, and divide decimals, fractions, and mixed numbers including rational numbers. You will also solve equations involving decimals and fractions. : CORE e,,* ." Chapter at a Glance Lesson Standards for Mathematical Content 3-1 Adding and Subtracting Decimals 3-2 Multiplying Decimals 3-3 Dividing Decimals 3-4 Solving Equations Containing Decimals 3-5 Adding and Subtracting Fractions 3-6 Multiplying Fractions and Mixed Numbers 3-7 Dividing Fractions and Mixed Numbers 3-8 Solving Equations Containing Fractions Problem Solving Connections Performance Task Assessment Readiness CC.7.NS.1, CC.7.NS.1b, CC.7.NS.1c, CC.7.NS.3 CC.7.NS.2, CC.7.NS.3 CC.7.NS.2, CC.7.NS.3 CC.7.NS.2, CC.7.EE.4 CC.7.NS.1b, CC.7.NS.1c CC.7.NS.2, CC.7.NS.2c, CC.7.NS.3 CC.7.NS.2, CC.7.NS.2b, CC.7.RP.1 CC.7.NS.2c, CC. 7 .EE.4

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Applying Rational Numbers

Chapter Focus You will add subtract multiply and divide decimals fractions and mixed numbers including rational numbers You will also solve equations involving decimals and fractions

~~~MON CORE

e Chapter at a Glance Lesson Standards for Mathematical Content

3-1 Adding and Subtracting Decimals

3-2 Multiplying Decimals

3-3 Dividing Decimals

3-4 Solving Equations Containing Decimals

3-5 Adding and Subtracting Fractions

3-6 Multiplying Fractions and Mixed Numbers

3-7 Dividing Fractions and Mixed Numbers

3-8 Solving Equations Containing Fractions

Problem Solving Connections

Performance Task

Assessment Readiness

CC7NS1 CC7NS1b CC7NS1c CC7NS3

CC7NS2 CC7NS3

CC7NS2 CC7NS3

CC7NS2 CC7EE4

CC7NS1b CC7NS1c

CC7NS2 CC7NS2c CC7NS3

CC7NS2 CC7NS2b CC7RP1

CC7NS2c CC 7 EE4

Unpacking the Standards Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter

What It Means to You

You will solve multi-step equations by identifying the operations involved and undoing them in the opposite order

EXAMPLE Yancey wants to buy a new snowboard that costs $39600 If she earns $825 per hour at work how many hours must she work to earn enough money to buy the snowboard

825h =396 825h = 396 Use the Division Property of825 825

Equality

h =48

Yancey must work 48 hours

What It Means to You

You will multiply and divide rational numbers

EXAMPLE In 1954 the Sunshine Skyway Bridge toll for a car was $175 In 2007 the toll was of the toll in 1954 What was the toll in 200n

Write the decimal as a fraction

Write the mixed number as an improper fraction

Simplify

Multiply numerators Multiply denominators

The Sunshine Skyway Bridge toll for a car was $100 in 2007

~Ki~~J~ ff2~~E CC7NS2a J~~st~nlIthat multiplication is

imiddot fr9T1 fractions to rational gtbyt~lIirng that operations ~9Iitf~~~tO$Qtisfy th~ properties of

lpel~~etlsfparticularly the distributive J(~dingto products such as (-1)

i bullbullbull land the rules for multiplying ~fgll~fluTIi1~rs In~erpret products of middot~~t~niltnlllnbers by describing real-world contexts

middotmiddotmiddotmiddotmiddoti~iweablJtary ~$~ribpti~pf()perty (Propiedad

~I~ffbq~ivaJ For aU real numbers a b ~ITdca(b+c) =ab + ae and arb e)

middotyeniilb--middotac

What It Means to You

You will solve real-world and mathematical problems involving the four operations with rational numbers

EXAMPLE Use the bar graph to determine how many times longer a $100 bill is expected to stay in circulation than a $1 bill

8

7 6 It 5 ~ 4

3

2

0 $1 $5 $10 $20 $50 $100

9 1 1 _ 9 37 2-172

=tmiddot~ Jg~

1 1

= ~or6

A$100 bill is expected to stay in circulation 6 times longer than a $1 bill

What It Means to You

You will multiply and divide positive and negative rational numbers

EXAMPLE 2-153

-15 ~ == 15 2 T3

= ~5 T31 10=-T

= -10

Write the whole number as a fraction

Simplify

Multiply numerators Multiply denominators

Key Vocabulary multiplicative inverse (inverso multiplicativo) One of two numbers whose product is 1 also called reciprocal

Multiplicative Inverse Property (Propiedad inversa de la multiplicaci6n) The product of a nonzero number and its reciprocal or multiplicative inverse is one

reciprocal (recfproco) One of two numbers whose product is 1 also called multiplicative inverse

Name______________ Class ______ Date __----i

Adding and Subtracting Decimals

Essential question How do you add and subtract decimals

Adding Decimals with the Same Sign

A~ Curtis is making trail mix He combines 06 pound of dark raisins and 08 pound ofgolden raisins How many pounds of raisins did Curtis use in alIT

06 Find 06 + 08 --I

0(1

Start at 06 on the number line -1 o 2

Move 1081 = 08 units to the left I right because the second addend is positive I negative

06 + 08 = ____

Curtis used a total of ____ pounds of raisins

y~ A scuba diver is at an elevation of -15 meters compared to sea level She descends an additional 25 meters What is her final elevation 0

Use a negative number to represent descending Find -15 + (-25) -1 I-SStart at -15 on the number line

Move 1-251 = 25 units up I down because the -2

second addend is positive I negative

-15 + (-25) = ___ -3

The scuba divers final elevation is meters -4

~jljlJ~~middot -5Find each sum Use a number line if needed

1a 52 + 46 = ___ 1b -04 + (-07) = ___ 1c -31 + (-16) = ___

1d Estimation Explain how you could use estimation to check your answer to Problem lc

Conjecture Make a conjecture about the sign of the sum when you add two decimals with the same sign

Chapter 3 97 lesson 1

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Unpacking the Standards Understanding the standards and the vocabulary terms in the standards will help you know exactly what you are expected to learn in this chapter

What It Means to You

You will solve multi-step equations by identifying the operations involved and undoing them in the opposite order

EXAMPLE Yancey wants to buy a new snowboard that costs $39600 If she earns $825 per hour at work how many hours must she work to earn enough money to buy the snowboard

825h =396 825h = 396 Use the Division Property of825 825

Equality

h =48

Yancey must work 48 hours

What It Means to You

You will multiply and divide rational numbers

EXAMPLE In 1954 the Sunshine Skyway Bridge toll for a car was $175 In 2007 the toll was of the toll in 1954 What was the toll in 200n

Write the decimal as a fraction

Write the mixed number as an improper fraction

Simplify

Multiply numerators Multiply denominators

The Sunshine Skyway Bridge toll for a car was $100 in 2007

~Ki~~J~ ff2~~E CC7NS2a J~~st~nlIthat multiplication is

imiddot fr9T1 fractions to rational gtbyt~lIirng that operations ~9Iitf~~~tO$Qtisfy th~ properties of

lpel~~etlsfparticularly the distributive J(~dingto products such as (-1)

i bullbullbull land the rules for multiplying ~fgll~fluTIi1~rs In~erpret products of middot~~t~niltnlllnbers by describing real-world contexts

middotmiddotmiddotmiddotmiddoti~iweablJtary ~$~ribpti~pf()perty (Propiedad

~I~ffbq~ivaJ For aU real numbers a b ~ITdca(b+c) =ab + ae and arb e)

middotyeniilb--middotac

What It Means to You

You will solve real-world and mathematical problems involving the four operations with rational numbers

EXAMPLE Use the bar graph to determine how many times longer a $100 bill is expected to stay in circulation than a $1 bill

8

7 6 It 5 ~ 4

3

2

0 $1 $5 $10 $20 $50 $100

9 1 1 _ 9 37 2-172

=tmiddot~ Jg~

1 1

= ~or6

A$100 bill is expected to stay in circulation 6 times longer than a $1 bill

What It Means to You

You will multiply and divide positive and negative rational numbers

EXAMPLE 2-153

-15 ~ == 15 2 T3

= ~5 T31 10=-T

= -10

Write the whole number as a fraction

Simplify

Multiply numerators Multiply denominators

Key Vocabulary multiplicative inverse (inverso multiplicativo) One of two numbers whose product is 1 also called reciprocal

Multiplicative Inverse Property (Propiedad inversa de la multiplicaci6n) The product of a nonzero number and its reciprocal or multiplicative inverse is one

reciprocal (recfproco) One of two numbers whose product is 1 also called multiplicative inverse

Name______________ Class ______ Date __----i

Adding and Subtracting Decimals

Essential question How do you add and subtract decimals

Adding Decimals with the Same Sign

A~ Curtis is making trail mix He combines 06 pound of dark raisins and 08 pound ofgolden raisins How many pounds of raisins did Curtis use in alIT

06 Find 06 + 08 --I

0(1

Start at 06 on the number line -1 o 2

Move 1081 = 08 units to the left I right because the second addend is positive I negative

06 + 08 = ____

Curtis used a total of ____ pounds of raisins

y~ A scuba diver is at an elevation of -15 meters compared to sea level She descends an additional 25 meters What is her final elevation 0

Use a negative number to represent descending Find -15 + (-25) -1 I-SStart at -15 on the number line

Move 1-251 = 25 units up I down because the -2

second addend is positive I negative

-15 + (-25) = ___ -3

The scuba divers final elevation is meters -4

~jljlJ~~middot -5Find each sum Use a number line if needed

1a 52 + 46 = ___ 1b -04 + (-07) = ___ 1c -31 + (-16) = ___

1d Estimation Explain how you could use estimation to check your answer to Problem lc

Conjecture Make a conjecture about the sign of the sum when you add two decimals with the same sign

Chapter 3 97 lesson 1

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

~Ki~~J~ ff2~~E CC7NS2a J~~st~nlIthat multiplication is

imiddot fr9T1 fractions to rational gtbyt~lIirng that operations ~9Iitf~~~tO$Qtisfy th~ properties of

lpel~~etlsfparticularly the distributive J(~dingto products such as (-1)

i bullbullbull land the rules for multiplying ~fgll~fluTIi1~rs In~erpret products of middot~~t~niltnlllnbers by describing real-world contexts

middotmiddotmiddotmiddotmiddoti~iweablJtary ~$~ribpti~pf()perty (Propiedad

~I~ffbq~ivaJ For aU real numbers a b ~ITdca(b+c) =ab + ae and arb e)

middotyeniilb--middotac

What It Means to You

You will solve real-world and mathematical problems involving the four operations with rational numbers

EXAMPLE Use the bar graph to determine how many times longer a $100 bill is expected to stay in circulation than a $1 bill

8

7 6 It 5 ~ 4

3

2

0 $1 $5 $10 $20 $50 $100

9 1 1 _ 9 37 2-172

=tmiddot~ Jg~

1 1

= ~or6

A$100 bill is expected to stay in circulation 6 times longer than a $1 bill

What It Means to You

You will multiply and divide positive and negative rational numbers

EXAMPLE 2-153

-15 ~ == 15 2 T3

= ~5 T31 10=-T

= -10

Write the whole number as a fraction

Simplify

Multiply numerators Multiply denominators

Key Vocabulary multiplicative inverse (inverso multiplicativo) One of two numbers whose product is 1 also called reciprocal

Multiplicative Inverse Property (Propiedad inversa de la multiplicaci6n) The product of a nonzero number and its reciprocal or multiplicative inverse is one

reciprocal (recfproco) One of two numbers whose product is 1 also called multiplicative inverse

Name______________ Class ______ Date __----i

Adding and Subtracting Decimals

Essential question How do you add and subtract decimals

Adding Decimals with the Same Sign

A~ Curtis is making trail mix He combines 06 pound of dark raisins and 08 pound ofgolden raisins How many pounds of raisins did Curtis use in alIT

06 Find 06 + 08 --I

0(1

Start at 06 on the number line -1 o 2

Move 1081 = 08 units to the left I right because the second addend is positive I negative

06 + 08 = ____

Curtis used a total of ____ pounds of raisins

y~ A scuba diver is at an elevation of -15 meters compared to sea level She descends an additional 25 meters What is her final elevation 0

Use a negative number to represent descending Find -15 + (-25) -1 I-SStart at -15 on the number line

Move 1-251 = 25 units up I down because the -2

second addend is positive I negative

-15 + (-25) = ___ -3

The scuba divers final elevation is meters -4

~jljlJ~~middot -5Find each sum Use a number line if needed

1a 52 + 46 = ___ 1b -04 + (-07) = ___ 1c -31 + (-16) = ___

1d Estimation Explain how you could use estimation to check your answer to Problem lc

Conjecture Make a conjecture about the sign of the sum when you add two decimals with the same sign

Chapter 3 97 lesson 1

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Key Vocabulary multiplicative inverse (inverso multiplicativo) One of two numbers whose product is 1 also called reciprocal

Multiplicative Inverse Property (Propiedad inversa de la multiplicaci6n) The product of a nonzero number and its reciprocal or multiplicative inverse is one

reciprocal (recfproco) One of two numbers whose product is 1 also called multiplicative inverse

Name______________ Class ______ Date __----i

Adding and Subtracting Decimals

Essential question How do you add and subtract decimals

Adding Decimals with the Same Sign

A~ Curtis is making trail mix He combines 06 pound of dark raisins and 08 pound ofgolden raisins How many pounds of raisins did Curtis use in alIT

06 Find 06 + 08 --I

0(1

Start at 06 on the number line -1 o 2

Move 1081 = 08 units to the left I right because the second addend is positive I negative

06 + 08 = ____

Curtis used a total of ____ pounds of raisins

y~ A scuba diver is at an elevation of -15 meters compared to sea level She descends an additional 25 meters What is her final elevation 0

Use a negative number to represent descending Find -15 + (-25) -1 I-SStart at -15 on the number line

Move 1-251 = 25 units up I down because the -2

second addend is positive I negative

-15 + (-25) = ___ -3

The scuba divers final elevation is meters -4

~jljlJ~~middot -5Find each sum Use a number line if needed

1a 52 + 46 = ___ 1b -04 + (-07) = ___ 1c -31 + (-16) = ___

1d Estimation Explain how you could use estimation to check your answer to Problem lc

Conjecture Make a conjecture about the sign of the sum when you add two decimals with the same sign

Chapter 3 97 lesson 1

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name______________ Class ______ Date __----i

Adding and Subtracting Decimals

Essential question How do you add and subtract decimals

Adding Decimals with the Same Sign

A~ Curtis is making trail mix He combines 06 pound of dark raisins and 08 pound ofgolden raisins How many pounds of raisins did Curtis use in alIT

06 Find 06 + 08 --I

0(1

Start at 06 on the number line -1 o 2

Move 1081 = 08 units to the left I right because the second addend is positive I negative

06 + 08 = ____

Curtis used a total of ____ pounds of raisins

y~ A scuba diver is at an elevation of -15 meters compared to sea level She descends an additional 25 meters What is her final elevation 0

Use a negative number to represent descending Find -15 + (-25) -1 I-SStart at -15 on the number line

Move 1-251 = 25 units up I down because the -2

second addend is positive I negative

-15 + (-25) = ___ -3

The scuba divers final elevation is meters -4

~jljlJ~~middot -5Find each sum Use a number line if needed

1a 52 + 46 = ___ 1b -04 + (-07) = ___ 1c -31 + (-16) = ___

1d Estimation Explain how you could use estimation to check your answer to Problem lc

Conjecture Make a conjecture about the sign of the sum when you add two decimals with the same sign

Chapter 3 97 lesson 1

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

~~li~~1Y~ Adding Decimals with Different Signs

In January the weight of a lion at a zoo changed by 18 pounds In February its weight changed by -23 pounds What was the overall increase or decrease in the lions weight

Find 18 + (-23)

Start at 18 (I -2

I I I I I -1

I I I I I 0

1 I I 1 I I I I 1)0 2

Move 1-231 = 23 units to the ~______ because the second addend is ______

The result is ____

Overall the lions weight increased decreased by ____ pound

Ernesto writes a check for $250 Then he deposits $6 in his checking account What is the overall increase or decrease in the accounts balance

Use a positive number to represent a deposit and a negative number to represent a withdrawal or a check

Find -25 + 6

Start at -25 ~I Imiddot -s -4 -3 -2 -1 0 1 2 3 4 S

Move 161 = 6unitstothe ______

because the second addend is ______

The result is ____

The account balance increases decreases by $ ____

Find each sum Use a number line if needed

2a -96 + 64 =___ 2b 48+(-21)= ___ 2e 7 + (-74) = ___

Conjecture Make a conjecture about the sign of the sum when you add two decimals with different signs

Error Analysis Martin says that the sum of341 and -152 is -189 Without performing the addition how can you tell that Martins answer is incorrect

Chapter 3 98 Lesson 1

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

~ c E 8 enc

~ c J 0shy

1 J

J S E ~ c

~ J o

J

subtract a positive decimal move to the left on the number line a negative decimal move to the right

Subtracting Decimals

At the beginning of summer the depth of water in a pond was 13 meters At the end of summer the depth was 07 meter What was the change in the depth of the water

Subtract the beginning depth from the ending depth to find the change Find 07 -13 07 _I

~~I~I~I~I~1~1~1~1-41-41-+1-+1-+1-+1-+1-+-+I~1~I~I~I- Start at 07 -2 -1 0 2

Move 1131 = 13 units to the left right because you are subtracting a positive negative decimal

The result is ____

The depth of the water increased decreased by ___ meter

illY tillS ~ __0

Find each difference Use a number line if needed

28-(-16) ___ 3b -35 - 2 ___ 3c ~4-(-07) ___

bullbull~ bull Writing and Simplifying an Expression

At the beginning of an experiment the temperature in a test tube was -15 cc At the end of the experiment the temperature was -24 cc What was the change in temperature in the test tube

Think The change in temperature is

Start at ____

Move 1-151 = 15 units to the ______ because you are subtracting a decimal

-I 1 1 The result is ____ -3

1 1 1 -2

1 1 1 1 -1

1 1 1 1 1 1 1 1

The change is temperature was ____ degC

REFLECT

4 Explain why it makes sense that your answer has the sign it does

Chapter 3 99 Lesson 1

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Find each sum or difference Use a number line if needed

1 35 + (-29) ___ 2 -4 + (-25) ___ 3 -12 + 11 ____

427 3 ____ 5 -06 - 06 ____ 6 32 - (-03) ___

The table shows how the price of a stock changed over a 3-day period For 7-9 solve by writing and simplifying an expression Use a number line if needed

7 What was the total change in the price of the stock on Monday and Tuesday

Monday -080

Tuesday -030

Wednesday +120 8 The price of the stock at the beginning of the day

on Monday was $990 What was the price at the end of the day on Monday

9 At the beginning of the day on Thursday the price of the stock was $1000 At the end of the day on Thursday the price of the stock was $930 What was the change in the price of the stock on Thursday

10 Gerald had $2050 saved at the beginning of July During July the amount of his savings changed by -$550 Explain how you could use a number line to find the amount of Geralds savings at the end oOuly Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -14 - 23 Then simplify the expression and tell what the answer means in the context of the situation

12 Reasoning The sum of 85 and a number n is negative What are two facts you can conclude about the number n

Chapter 3 100 Lesson 1

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ___________ Class _____ Date ____

_MfliJ~ffPaji(~7-

Add Estimate to check whether each answer is reasonable

1 614 + 891 2 451 + 1308 3 1254 + 2108

4 3422 + (-185) 5 -1010 + (-59) 6 687 + (-316)

79+568 8 -1551 + 855 9 3636 + 5454

Subtract

10 623 - 362 11867 - 687 12 2894 - 948

13 2357 - 684 14 1661 - 756 15 3208 - 1237

16 19 - 692 17 42 - 3189 1823 - 2145

19 462 - 027 20 22 - 1863 21 589 - 2958

22 Anna swims the length of the pool in 3845 seconds and then swims the length of the pool again in 4238 seconds What is her total time for 2 lengths of the pool

23 Po has 2 gerbils named Yip and Yap Yip weighs 362 ounces and Yap weighs 279 ounces How much heavier is Yip than Yap

Chapter 3 101 Practice and Problem Solving

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 In the Pacific Ocean the Philippine Trench is 1005 kilometers deep In the Atlantic Ocean the Brazil Basin is 612 kilometers deep How much deeper is the Philippine Trench than the Brazil Basin

3 A team of mountain climbers makes camp 148 kilometers above sea level They climb another 291 kilometers to the mountains peak How tall is the mountain

Choose the letter for the best answer

This table gives the heights of the tallest mountains on each of the seven continents

The Seven Summits Height I

Mountain Country (km) i Mt Everest Nepal-Tibet 885 i

Mt Aconcagua Argentina 696 I I I United I

I Mt McKinley States 619 I

I Mt Kilimanjaro Tanzania 590 I

Mt Elbrus Russia 564 i

Vinson Massif Antarctica 490

Puncak Jaya New Guinea 488 I

Chapter 3 102

2 Hawaiis Mauna Kea measures 975 kilometers from its base to its peak The base of Mauna Kea lies 555 kilometers below the ocean What is the height of the part of Mauna Kea that is above sea level

4 At dawn the temperature at the summit of a mountain was -85 degG By noon the temperature had increased 36 degG What was the temperature at noon

5 In 2000 Joby Ogwyn became the youngest person to climb each of the seven summits How much higher did he climb on Vinson Massif than on Puncak Jaya

A 002 km G 018 km

B 012 km D 978 km

6 Mt Kilimanjaro Mt Elbrus and Mt Aconcagua were the first three of the seven summits Joby climbed What was the total height he climbed

F 162 km H 185 km

G 175 km J 22 km

7 Mt Aconcagua is about 13 kilometers taller than which mountain

A Mt Everest

B Mt Elbrus

G Mt McKinley

D Vinson Massif

J

Practice and Problem Solving

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name__________________ _______ Date ___

Multiplying Decimals

Essential question How do you multiply decimals

You can use base-ten blocks to model decimals Always combine base-ten blocks into the largest group possible For example ten 0001 blocks should be combined into one 001 block

= 001 = 0001= 01

_aiin-IOE X AMP LE Multiplying Decimals by Whole Numbers

A A package of sunflower seeds has a mass of 01 kilogram What is the mass of 3 packages of sunflower seeds

Find 301

Multiplication is repeated addition so 301 = 01 + 01 + 01

01 +01 +01 =___ 301 = ____

The mass of 3 packages of sunflower seeds is ____ kilogram

B A ring has 4 diamonds each with a mass of 003 carat What is the total mass of the diamonds in the ring

Find 4 bull 003

4003 = 003 + 003 + 003 + 003 III 111111 IIICombine ten 001 blocks into one

003 003 00301 block 003

01 + 002 = ____

4003 = _____

The total mass of diamonds in the ring is ____ carat

TRY THIS 01 002

Use base-ten blocks to find each product

1a 406 = 1b 9002 =____ 1c 2 026 =____

=1

01 01 01

Chapter 3 103 Lesson 2

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

1d How many 01 blocks do you need to combine to form a 1 block What multiplication statement are you modeling when you do this Explain

1e When you multiply a whole number greater than 1 and a positive decimal is the product less than or greater than the decimal Explain how you know

cgtltgtCo~W=

Multiplying Decimals by Decimals

A recycling center charges customers $040 per pound to recycle electronic equipment How much would the center charge to recycle a broken camera that weighs 07 pound

Use a decimal grid to find 0407

Shade 04 Shade 07 The area where the shaded horizontally vertically regions overlap is the product

I I I

i

I L

I

I

x I

=

shy

I I

I I

04 x 07 = The recycling center would charge ____ to recycle the camera

Use decimal grids to find each product

la 0707 = ____ lb 03 08 = ____ le 05 02 =____

ld When you use a decimal grid to find a product of two positive decimals less than I does it matter which factor you shade horizontally and which factor you shade vertically Explain

Chapter 3 104 Lesson 2

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

2e Explain how you could use two decimal grids to find 12 bull 03

2f When you multiply two positive decimals less than 1 is the product less than or greater than I Explain how you know

1JlIIIIiiAtJ 5~~J~r~~ Analyzing Decimal Multiplication

Amy found 03 bull 08 and got an answer of 086 Her work is shown below

5hade 03 horizontally_ 5hade 08 vertically There are 8 squares Shaded n all 50 03 bull 08 =08

3a Identify the error that Amy made Explain your reasoning

3b What is the correct answer to the problem 03 bull 08

3c Estimation How could Amy have used estimation to determine that her answer is incorrect

Chapter 3 105 Lesson 2

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Find each product Use base-ten blocks or decimal grids if needed

1 00026 = ____ 2 8004 = ____ 3 5016 = ____

4 0901 = ____ 5 0608 6 14 bull 04 = ____

The bar graph shows the top speeds of four animals Using the table for 7-9 write and simplify an expression to solve the problem Use base-ten blocks or decimal grids if needed (Hint Multiply the time by an animals top speed to find the distance it can run in that time)

7 At top speed how far can a grizzly bear run in 3 minutes

8 At top speed how far can a pronghorn antelope run in 05 minute

9 Which animal can run farther a coyote running at top speed for 03 minute or a jackrabbit running at top speed for 02 minute Support your answer

Top Speeds

08 ~ 06 05 s 04 -c 03 IIgt 02 g- 01 I- 0

1~~~~~-~middot-~middot~~middot~middot~-middot--~---middot~---middotmiddotmiddot

O9 I--- ----~---~- -~~-

071--middot_middotmiddotmiddotmiddotmiddotmiddotmiddotmiddotmiddot-_middot_middotmiddot_middotmiddotmiddot

E Q) 00 co01shycll o c tt ltll

10 Trevor bought 3 erasers for $046 each Explain how you could use base-ten blocks to find the total amount Trevor spent

11 The variable n represents a positive decimal If the product 3 bull n is less than 3 what is a possible value of n Support your answer

Chapter 3 106 Lesson 2

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ___________ Class _____ Date ____ c

Multiply

1 6 - 03 2 3 - 005 307-4

4 8 - 61 574-6 6 14 - 9

7 48 - 7 8 3 - 82 9 55 - 8

10 15 - 6 1179-2 12 5 - 69

Multiply Estimate to check whether each answer is reasonable

13 63 - 78 14 97middot (-47) 15 68middot 09

16 28 - 82 17 -7 - 642 18 19 - 722

19 -53 bull (-84) 20 716 - 003 21 156middot (-78)

2246 - 31 23 062 bull 145 24 -574 bull 19

25 Jordan jogged 48 miles each day for 21 days last month How many miles did she jog last month

Chapter 3 107 Practice and Problem Solving

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 A group of 6 adults bought tickets to a play at the community center The tickets cost $1750 each How much did the tickets cost in all

3 Movie tickets for senior citizens cost 08 times as much as regular adult tickets Adult tickets cost $750 How much do senior citizen tickets cost

Choose the letter for the best answer

5 The population of Illinois in 2000 was about 157 times its population in 1940 If the population of Illinois in 1940 was about 79 million what was its population io 2000 to the nearest tenth of a million

A 12 million

B 95 million

C 124 million

D 158 million

7 Nauru and Gibraltar are among the smallest countries in the world Nauru has about 328 times the area that Gibraltar does Gibraltar is 25 square miles What is the area of Nauru A 078 mi2

B 082 mi2

C 578 mi2

D 82 mi2

--

2 Student tickets for the basketball playoffs cost $925 each How much do 9 student tickets cost

4 About 305 million Americans attend classical music concerts The average concertgoer attends 29 concerts per year About how many tickets to classical concerts are sold each year

6 In 2004 the United States had 325 million broadband subscribers This number is expected to increase 175 times by 2008 How many broadband subscribers is the United States expected to have in 2008

F 24375 million

G 3425 million

H 56875 million

J 56875 million

8 From 1999 to 2008 the United States Mint is issuing a series of 50 quarters representing each of the 50 states What is the cost of collecting 10 of each quarter

F $100

G $125

H $150

J $250

Chapter 3 108 Practice and Problem Solving

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

r===shy

Name______________ Class______ Date____

Dividing Decimals

Essential question How do you divide decimals

You can use base-ten blocks to model dividing decimals by whole numbers Remember

if

=001 =0001=1 =01

a~ Dividing Decimals by Whole Numbers

Lori bought 2 sheets of scrapbook paper for a total of $034 What is the price of each sheet if they are the same price

Use base-ten blocks to frod 034 + 2

First model 034 03 004

Separate one 01 block into ten 001 blocks

Then divide the blocks into 2 equal groups II(shown by the black rectangles)

In each of the two equal groups there are II____ 01 blockand ____ 001 blocks

01 + 007 = ___

034 -7- 2 = ____ 1111111 The price of each piece of scrapbook paper is

$-- shy 111111 TRY THIS

Use base-ten blocks to find each quotient

1a 09 -7- 3 =____ 1b 064 -7- 4 = ____ 1( 04 -7- 5 = ____

REfLECT

1d When you divide a positive decimal by a whole number greater than 1 is the quotient less than or greater than the decimal Explain how you know

Chapter 3 109 Lesson 3

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

You can use decimal grids to model dividing decimals by decimals

Dividing Decimals by Decimals

Antoine skated 08 mile One lap of the skating rink is 02 mile How many laps did Antoine skate

Use a decimal grid to find 08 + 02

Shade 8 columns to represent 08

Divide the 8 columns into groups that each contain 02 of a decimal grid or 2 columns

There are __groups that each contain 02 of a grid

08+02= __

Antoine skated __ laps

middot8middotmiddotmiddot A tablet computer weighs 13 pounds The weight of several tablet computers in a shipping box is 39 pounds How many tablet computers are in the shipping box

Use decimal grids to find 39 + 13

Shade 3 grids and 9 columns of a fourth grid to represent 39

Divide the shaded grids and columns into groups that each contain 13 decimal grids or 13 columns

There are __ groups thateach contain 13 decimal grids

39+ 13= __

There are __ tablet computers in the shipping box

Use decimal grids to find each quotient

06+03= __ 2b

When you use decimal grids to divide a decimal by a decimal how do you represent the dividend How do you represent the divisor

15 + 03 2c 24 + 12 = __

Chapter 3 110 Lesson 3

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

You can use what you have learned about multiplying and dividing decimals to solve problems involving rational numbers

a~ Multiplying and Dividing Rational Numbers

The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Round your answer to the nearest tenth

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) -45 -3 2 -15 -2 3 -5

0 ~

E 0

U 0gt

~ 0 gt shyt gt 0 ~

J E ~ 0 1 0gt gt 0 J

Average = _-45+-__+___+--__+--__+--__+-__

You use the order of operations to simplify

The first step is to add the numbers in the _ --------_

The second step is to by the number in the ________

Average = -- = ____

The average daily high temperature is approximately ____

TRY THIS

3a Marcy is climbing down a hill She descends 18 meters each minute for 5 minutes Write and simplify an expression to find Marcys total change in elevation

3b The table gives the daily high temperatures for one week during the winter Find the average daily high temperature for the week Show your work

Day Mon Tues Wed Thurs Fri Sat Sun

Temperature (OF) 64 23 0 -64 -23 5 -5

Chapter 3 111 Lesson 3

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Find each quotient Use base-ten blocks or decimal grids if needed

1 036- 4 ____ 2 02- 8 ____ 3 036- 6 ____

4 1- 02 ____ 5 07- 01 ____ 6 54- 18 ____

The grocery receipt shows how much Russell spent on fruit for a picnic Solve each problem by writing and simplifying an expression Use base-ten blocks or decimal grids if needed

7 How much do pears cost per pound (Hint Divide the amount Russell spent on pears by the number of pounds of pears he bought)

Pears 3 pounds

Apples 4 pounds

Oranges 2 pounds

Grapes 23 pounds

Total Sale

$240

$380

$256

$460

$1336

8 How much do grapes cost per pound

9 Which of the fruits Russell bought has the greatest price per pound Support your answer

10 The table shows how the temperature in a refrigerator changed over the past 5 hours Write and simplify an expression to find the average hourly change in temperature

I 0 c to y 0 J

~

~ r I

8 c l

c nr to 0 3

J lt

1 04

-22

3 16 4 -1

5 05 11 Ms Howe made 3 withdrawals of $780 each from her

bank account Write and simplify an expression to find the total change in her account balance

12 Explain how you knew which operation to use to solve Problem 11

Chapter 3 112 Lesson 3

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ____________ Class _____ Date ____

Divide

1 6 + 025 2 7874 + 127 3 7348 + -167

4 56525 + 085 5 4422 + (-67) 6 -646 + 004

Divide Estimate to check whether your answer is reasonable

7 63 + (-45) 8 8 + 32 9 87 + 725

gtc amp E 10 -36 + 16 1142 + 48 12 90 + 036 8 c

~ 0 gt

1

sect gt

tor c

~ l

~ 13 Freddie used 675 gallons of gas to drive f 15525 miles What was his cars gas mileage

14 The members of a book club met at a restaurant for dinner The total bill was $11295 and they shared the bill equally Each person paid $1255 How many members are there in the book club

Chapter 3 113 Practice and Problem Solving

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 Fran has a plank of wood 465 meters long She wants to cut it into pieces 085 meters long How many pieces of wood that length can she cut from the plank

3 Rhosanda is downloading a file from the Internet The size of the file is 745 MB The file is downloading at the rate of 0095 MB per second How many seconds will it take to download the entire file Round your answer to the nearest second

Choose the letter for the best answer

5 Mount McKinley in Alaska is about 3848 miles high If a mountain climber can climb 025 miles per day about how long to the nearest day would it take to climb Mount McKinley

A about 5 days

B about 8 days

C about 12 days

D about 15 days

7 In the 2000 Summer Olympics Michael Johnson ran the 400-meter race in 4384 seconds To the nearest hundredth what was his speed in meters per second

A 912 meters per second

B 1096 meters per second

C 109 meters per second

D 9124 meters per second

2 Jeremy has a box of 500 nails that weighs 135 kilograms He uses 60 nails to build a birdhouse How much do the nails in the birdhouse weigh

4 Sean has a piece of poster board with an area of 47628 square centimeters He cuts it into equalshysized squares each with an area of 3969 square centimeters How many squares can he cut from the piece of poster board

6 In 2000 a production worker in Japan who worked 385 hours in a week would have earned an average of $847 What was the hourly wage

F $220 per hour

G $320 per hour

H $2200 per hour

J $3200 per hour

8 In the 2000 Summer Olympics the United States relay team ran the 1 600-meter relay in 2 minutes 5635 seconds To the nearest hundredth what was the speed for the relay race in meters per second

F 283 meters per second

G 624 meters per second

H 907 meters per second

J 2839 meters per second

Chapter 3 114 Practice and Problem Solving

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

S

~ sect

E J o t

Name ______________ Class ______

Solving Equations Containing Decimals

Essential question How do you solve equations containing decimals

o~ Solving Addition and Subtraction Equations

Angelita bought a shirt on sale for $824 The shirt was discounted from the regular price by $125 The equation -125 =824 can be used to find the regular price of the shirt in dollars What is the regular price of the shirt

A Use an integer equation to model the problem

Round 125 to 1 and round 824 to 8 The decimal equation can be modeled by the integer equation r - 1 = 8

r - 1 = 8 1 is subtracted from r

+ + Add 1 to both sides to undo the subtraction

r = Solvefor r

B Solve the decimal equation

To solve the integer equation you added a number to both sides So to solve the decimal equation you will also add a number to both sides

r - 125 = 824 125 is subtracted from r

+ + Add ___ to both sides to undo the subtraction

r = Solve for r

The regular price of the shirt is $

REFLECT

1a How does the solution to the integer equation help to show that your solution to the decimal equation is correct

1b How is solving a decimal equation similar to solving an integer equation How are they different

Chapter 3 115 Lesson 4

01

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

A test tube has a mass of 104 grams Lena adds some salt to the test tube The combined mass of the test tube and salt is 158 grams The equation 104 + 5 = 158 can be used to find 5 the mass of the salt in grams

1c What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

1d How can you solve the decimal equation What is its solution

How many grams of salt did Lena add to the test tube __________

~w=c

Solving Multiplication and Division Equations

A candle maker places an order for wax The wax is priced at $180 per pound and his order costs $1170 The equation 18p = 117 can be used to find p the number of pounds of wax the candle maker ordered How many pounds did he order

A Use an integer equation to model the problem

Round 18 to 2 and round 117 to 12 The decimal equation can be modeled by the integer equation 2p = 12

2p 12 P is multiplied by 2

= Divide both sides by 2 to undo the multiplication

p = Solveforp

B Solve the decimal equation

To solve the integer equation you divided both sides by a number So to solve the decimal equation you will also divide both sides by a number

18p = 117 P is multiplied by 18

Divide both sides by _______

Solveforpp

The candle maker ordered _______ pounds of wax

Chapter 3 116 Lesson 4

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

18yl1KIS gt gt-- ---~ - --

Ms Joyners car can travel 304 miles per gallon of gasoline The equation ~ = 304 can be used to find m the number of miles Ms Joyner can drive on 45 gallons of gasoline

2a What integer equation can you use to model the problem How can you solve the integer equation and what is its solution

2b How can you solve the decimal equation What is its solution

How far can Ms Joyner drive on 45 gallons of gasoline ____________

B middot1I31 ~~~~M P L E Writing and Solving Decimal Equations

Julios time in the first half of a race is 5806 seconds What time t does Julio need to get in the second half of the race to tie his best race time of 11194 seconds Write and solve an equation to find the value of t

Think

5806 11194

5806 + t = Write the equation

Subtract -----from both sides

t = Solve for t

Julio needs to get a time of _____ seconds in the second half of the race to tie his best race time

REFLEC1

3a Explain how you could use substitution to check your answer to e

TRY THIS

Estella bought 7 guitar strings for a total of $2520 Each string costs the same amount Write and solve an equation to find c the cost in dollars of each string

Chapter 3 117 Lesson 4

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Solve

1 86 = n + 33 2 b - 65 = 24 3 x + 04 = -03

4 48m 1728 5 O~2 = -16 6 42 =08k

Use the breakfast menu to solve each problem

7 Mr Delaney bought muffins for his family He paid a total of $650 Write and solve an equation to find m the number of muffins Mr Delaney bought

Biscuit $085

Omelet $495

Pancakes $485

Muffin $130

Fresh fruit $225

Orange juice $185

8 Roslyn has $475 to spend on breakfast She buys a glass of orange juice Write and solve an equation to find a the amount Roslyn has left to spend on food

9 James had a coupon for $050 off his order He ordered one item off the menu used the coupon and paid $435 Which item did James order Write and solve an equation to support your answer

10 Error Analysis Carly solved the equation = 6 as shown at right What error did she make What is the correct solution of the equation

11 Is the solution of the equation x - 23 = 18 less than 18 or greater than 18 Explain how you know without solving the equation

Chapter 3 118 Lesson 4

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ___________ Class _____ Date ____ i

_Ad~itif)nal Practice) Solve

1 t+O77=93 2 p-134=-118 3 r+214=78

4 365 + e = -14 5 w-167 = 827 6 z-172=713

7 P - 675 = 2481 8 h + 269 =1274 9 k + 892 = -4762

10 x - 045 = 597 11 108 + n =1572 12 Y - 632 =073

13 43p = 2881 14 77j = 7623 1538g=-10412

16 1836 =27y 17 9996 =68x 18 29392 = 176c

e f d 19 74 =69 20 127 = 156 21 97 = 208

w m s 22 -02 = 154 23 98 =17 24 1435 =-52

25 Jeff paid a flat fee of $26950 for a years worth of vet visits for his 4 cats He made 14 visits during the year What was the average cost per visit

Chapter 3 119 Practice and Problem Solving

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 The diameter of the secondary mirror in NASAs Hubble telescope is 03 meter The primary mirror is 8 times as large What is the diameter of the primary mirror

3 Brianna drives 335 miles to work every day This is 175 miles less than Darius drives to work every day How far does Darius drive to work

Choose the letter for the best answer

The table shows the orbital velocity of some of the planets

5 How much greater is the orbital velocity of Mercury than that of Uranus

A 3397 miles per second

B 2551 miles per second

C 703 miles per second

D 423 miles per second

6 How many miles does Mercury travel in an hour

F 17844 miles

G 107064 miles

H 107064 miles

J 17844 miles

7 How many miles does Jupiter travel in a minute

A 812 miles

B 812 miles

C 4872 miles

D 4872 miles

2 A cubic centimeter of titanium weighs 4507 grams The same volume of gold weighs 193 grams How much more does a cubic centimeter of gold weigh

4 The weight of an object on Mars is 038 times its weight on Earth How much would a 125-pound person weigh on Mars

PI t 0 bt I V I tane s r I a eocl~

Orbital Velocity Planet (mils)J Mercury 2974

Venus 2176

Earth 185 i Mars 1499

I J t upler 812

Saturn 602

423I Uranus

8 During the time it takes Saturn to travel 32508 miles how much time has elapsed on Earth

F 5400 minutes

G 175719 minutes

H 195698 seconds

J 90 minutes

Chapter 3 120 Practice and Problem Solving

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

gtshy

c 0 E 0 v Olc c il j5 =gtshy1

sect =gt

r so E ~ c 0 +c Ol =gt 0r

Name______________ Class ______ Date____

Adding and Subtracting Fractions

Essential question How can you add rational numbers

middotmiddotmiddotmiddot~tli ~E X AMP l Emiddot Adding Fractions with the Same Signa

~~~

A Alisha walked l mile to the library and then i mile to the post office What is the total distance Alisha walked

Find i + ~ Start at i on the number line I~ o 1

Move I~I =~ unit to the right because the 2

second addend is positive

The result is ____ Alisha walked a total of ____ mile

B Kyle pours out ~ cup of milk from a pitcher Then he pours out another cup from the pitcher How many cups of milk does he pour out altogether

Use negative numbers to represent amounts that are poured out of the pitcher

0(1 1 ) Find -~ + ( -~) Start at -l -2 -1 0

Move I-~I = unit to the left because the second addend is negative

The result is

Kyle pours out cups

TRVTHiS

Use a number line to find each sum

1a 312+ 112 = 1b -2~ + (-4~) =

0(1 I~ 1 110 0 2 3 4 5 -7 -6 -5 -4 -3 -2 -1 0

REFLECT

1c Estimation Explain how you could use estimation to check your answer to Problem lb

Chapter 3 121 Lesson 5

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

add fractions with the same sign find the sum of their absolute values use the sign of the fractions

Adding Fractions with Different Signs

A baby panda bear gained ~ pound Then it lost1pound What is the total change in the pandas weight

Use a negative number to represent a loss of weight

Find ~ + ( - k) Start at~ 1)0

Move I-kl kunit to the ----- shyo 1

2

because the second addend is ______

The result is ______

The total change in the pandas weight is ______ pound

In the first half of the year a citys rainfall was ~~ inch below normal In the second half of the year the citys rainfall was ~ inch above normal What was the citys overall rainfall compared to normal for the year

Use a negative number to represent an amount below normal

dIll 11Fill -16 +4 Start at -16 1

-1 o 1Move II unit to the _____ 2

because the second addend is ______

The result is ______

The overall rainfall is ______ inch below above normal

Use a number line to find each sum

7 shy2a +10- ____ 2b ~ + (-~) =

1)0II -1 0 1

2-1 o

Chapter 3 122 Lesson 5

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

lt~pound~~~CT

9 32e Error Analysis Aaron says that the sum of -16 and 16 is Without performing the addition how can you tell that Aarons answer is incorrect

o add fractions with different signs find the difference of their absolute values and use the sign of the fraction with the greater absolute value

MBiitiia E XA MP L E Writing and Simplifying an Expression bull

gtc ro 0shyS 3 01c

~ a gt shyt gt

~ t 5 it ~ c 0s 01 gt 0 t

Compared to sea level an explorer in a cave is at an elevation of -i mile She descends an additional i mile What is the explorers final elevation compared to sea level

Think The final elevation is

the starting elevation the change in efevation

Start at _____

Move I-~I = ~ unit to the ------ o 1 2

because the second addend is ______

The result is

The explorers final elevation is ___mileS above sea level

3a What if What if the explorer had gone up ~ mile instead of going down an additional ~ mile Explain how you could use mental math to find the explorers final elevation compared to sea level

Chapter 3 123 Lesson 5

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Find each sum Use a number line if needed

1 ~+~=_____ 2 +(-i)=-shy

5 14 -196 +~=_____ 5 -16+2=----- 6 ~+ (-2) =--shy

A restaurant has a large iced tea container The table shows the number of gallons of iced tea added to and used from the container each hour during lunch Solve each problem by writing and simplifying an expression

7 What was the overall change in the amount of iced tea in the container from 1100 to 1200

3 711 00-1200 4 8 8 What was the total amount of iced tea used

during lunch 1200-100 7 8

15 16

100-200 1 2

5 16

9 At 100 the container held 3~ gallons of iced tea How many gallons did the container hold at 2001

10 The temperature inside an airplane decreased 3~ OF during the first 20 minutes of a flight and then increased 2 OF during the next 20 minutes Explain how you could use a number line to find the total change in temperature Then find this amount

11 Give an example of a real-world situation that could be represented by the expression -~ + ( -l) Then simplify the expression and tell what the answer means in the context of the situation

Chapter 3 124 Lesson 5

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name____________ Class _____ Date ____ 3-5

I Practice

Add or subtract Write each answer in simplest form

1 2 4 8 7 51 - +- 2 +- 3 -- shy

5 5 15 15 12 12

9 7 7 11 2 64 --- 5 6 - +shy

10 10 12 12 7 7

3 1 8 57 + ~ 8 --- 9 -+shy15 15 16 16 21 21

4 3 3 110 - - - 11 - + - 12 3 - 3

5 4 8 2 5 25

7 5 9 513 + ~ 14 - -- 15 - +shy12 6 8 12 10 6

2 7 3 816 - - - 17 5 + 18 - - shy5 8 6 15 4 15

19 The school track is i mile in length Sherri ran 3 mile How much 8 3

farther does she have to go to get all the way around the track

20 The Millers budget ~ of their income for fixed expenses and ~ 2 8

of their income for savings What fraction of their income is left

Chapter 3 125 Practice and Problem Solving

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 During the recycling drive of the

material collected was bottles and -i was paper Cardboard boxes made up

1~ of the material How much of the

total do these three items represent

3 The school environmental club made a poster to celebrate Earth Day The

poster is ~ yard long and ~ yard

wide What is the difference in the length and width of the poster

Choose the letter for the best answer

5 The Reeds budget ~ of their income

1for rent and 4 for food How much of

their budget is left

5A 1 C 122

7B 3 D 124

7 Wong has 2 boxes of saltwater taffy

One box contains pound and the

7 other box contains 10 pound How

much taffy does he have all together

A 9

10 pound C 1

12 pounds

9 13 B 1 20 pounds D 1 20 pounds

Chapter 3 126

2 Decorations for school dances take i of the student councils budget

3Entertainment takes 10 of the budget

What fraction of the budget is left

4 Three students ran for president of the student council Eddie received

i of the votes Tamara received ~ of the votes Levi received the rest Which student won the election

6 Jasmines CD collection is ~ jazz

~ rap and the rest rock music What

fraction of her CDs is rock music

8 In 1992 about 100 people voted for

F 1 H 1 4 2

G 3 J 5 8 8

43

Bill Clinton for President About ~ voted for Ross Perot and the rest voted for George Bush About how many voted for George Bush

F 37

about 100

G 52

about 100

57 H about 100

3 J about 5

Practice and Problem Solving

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

--------------------------------------

Name______________ Class______ Date___

Multiplying Fractions and Mixed Numbers

Essential question How do you multiply fractions and mixed numbers

Modeling Fraction Multiplication

Mia has tpint of yogurt She eats i of it at lunch The expression i middott gives the number of pints Mia ate Use a grid to model i middotf-

A First model ~ to represent ~ pint of yogurt

The denominator is 2 so divide the square grid vertically into 2 equal parts

The numerator is I so shade 1 of the 2 vertical parts

B Next model t to represent the fraction that Mia ate

The denominator is 4 so divide the grid horizontally into 4 equal parts

The numerator is 3 Use a different color to shade 3 of the 4 horizontal parts

C The area where the two colors overlap represents f of ~ pint or the number of pints Mia ate

The grid is divided into a total of ____ parts

The number of parts in the area where the two colors overlap is ____

3 1 +-shy number of overlapping parts 4 bull2 =8 +-shy total number of parts

Mia ate ____ pint of yogurt

TRY THIS

Use a grid to find each product

1b 2 1 --=---shy3 3 1c 3 1--=---shy4 3

REFLECT

1 d Explain how the number of columns and the number of rows in the grid are related to the denominators of the fractions in O

_

Chaoter 3 127 Lesson 6

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

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U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

1e Are ~~ and ~~modeled the same way Explain

Modeling Multiplication of Mixed Numbers

Duane needs to run 11 miles for track practice So far he has run ~ of this

distance The expression ~ 11 gives the distance in miles Duane has run

so far Use grids to model ~ 11shyA First model 1 t to represent It miles

The denominator of the fraction is 4 so divide each grid vertically into 4 equal parts

Shade 1 entire grid to represent the whole number 1 Then shade 1 of the 4 vertical parts of the second grid to represent the fraction l

Next model ~ to represent the fraction of the distance Duane has run

The denominator is 3 so divide each grid horizontally into 3 equal parts

The numerator is 2 Use a different color to shade 2 of the 3 horizontal parts of each grid

e The area where the two colors overlap represents ~ of Iimiles or the distance Duane has run so far

Each grid is divided into a total of ____ parts

The total number of parts where the two colors overlap is ____

oE-- total number of overlapping parts

oE-- number of parts in one grid

So far Duane has run ____ mile which simplifies to ____ mile

Use a grid to find each product

2a 211= ____ 2b I 1 = ____ 2c 12~ shy24- 4 3 - 4 3---- shy

Chapter 3 128 Lesson 6

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

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U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

-----------

REflECT

2d Explain how you can use the model you drew in A to write 1 ~ as an improper fraction

You can also use number lines to help you solve problems involving multiplication of fractions and mixed numbers

Multiplying Rational Numbers

A Gina hiked down a mountain in ~-mile sections She hiked 4 sections How many miles did she hike down

Find (-~)4

You can use the Commutative Property of Multiplication

to write ( - ~)4 as ______

Move l unit to the left 4 times

The result is ____ I~ -3 -2 -1 o

Gina hiked down _____ miles

B Each batch of muffins requires 1 ~ cups of applesauce Sid needs to make 3 batches of muffins How many cups of applesauce does he need

Find (lj)3 I o 2 3 4 5 6

You can use the Commutative Property of Multiplication to

write (lj) 3as _____

Move 1j units to the ____ 3 times

The result is _____ Sid needs ____ cups of applesauce

TRY THIS

Use a number line to find each product

3a 3(~) = _____

Chapter 3 129 Lesson 6

Dc EX AMP L E

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

3d Error Analysis Kristie says that the product of 2 and Ii is IIWithout perfonning the multiplication how can you tell that Kristies answer is incorrect

5 ~5~5 Solving Multi-Step Problems

A chef adds i cup of stock to a stew 5 times while it is cooking During this time he also removes ~ cup of stew 4 times to serve to guests Write and simplify an expression to find by how much the overall amount of stew increased or decreased

Write an expression for the situation

() I t t

Number of Amount Removal of Number of -Amount times stock added stew times stew removed

added removed

Use the order of operations to simplify the expression

Multiply

Subtract

The expression simplifies to ___

The overall amount of stew increased decreased by ___

4a A chef adds 1cup of stock to a soup 7 times while it is cooking During this time he removes ~ cup of soup 3 times to taste it Write and simplify an expression to

find how much the overall amount of soup increased or decreased

How could you solve problem 4a without using multiplication

Chapter 3 130 Lesson 6

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE Use a grid or grids to find each product

2 l~ = 3 ~~= 4 4 ----- shy 4 3

~ 6 11~ =4 111 = bull 2 3 3 3

Use a number line to find each product

7 (-i)3 = ----- shy-3 -2 -1 o

o 2 3 4

9 (~)4=--_-- L I o 2

For 10-12 solve by writing and simplifying an expression

10 Nicholas has ~ pound of walnuts He adds 1of the walnuts to a bowl of pancake mix How many pounds of walnuts did Nicholas add to the bowl

11 A pitcher holds 41 cups of lemonade Daisy drank tof the lemonade How many cups of lemonade did Daisy drink

12 A hot-air balloon descends 4feet each minute for 5 minutes What is the total change in the elevation of the balloon

Chapter 3 131 Lesson 6

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

The table shows the amount of paint the drama club has for painting backgrounds For 13-15 solve by writing and simplifying an expression

13 Tobias used ~ of the blue paint What is the change in the amount of blue paint that the club has Blue 2

Green

Brown

14 Janna mixes ~ of the white paint with ~ of the green paint to make light green paint How many gallons oflight green paint does Janna make

15 Three students each use ~ gallon of brown paint Then they buy 2 more gallons of brown paint How much brown paint does the club have now

16 Describe how you could check your answer to problem 15

17 Explain how you could use the Associative Property of Multiplication to help you find the product (~ bull2 ) ~ by using mental math

18 Make a Conjecture When you multiply two positive fractions less than 1 is the product less than 1 or greater than 1 Explain

Chapter 3 132 Lesson 6

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ____________ Class _____ ----ciJj

Additional Practice 7-Multiply Write each answer in simplest form

115- shy

2 3

29- shy4

2 3 6 --shy

5

9 5 9 7 7 6 4 -

15 -shy7 5 -

14 --shy

9 6 --shy12 14

37 -12 - shy

7

58 15 - shy

6 9 21shy

3 -8

1 310 2- - shy

3 5

2 1113--shy

5 2 5 2

124--shy6 5

2 213 2- - shy

5 3

314 3- shy

4 2 -5

1 315 8-- shy

6 7

1 316 2- - 3shy

3 8 17 1

3 5

- 6~3

2 518 2-- 4shy

5 6

19 Rolf spent 15 hours last week practicing his saxophone If ~ of 10

the time was spent practicing warm-up routines how much time did he spend practicing warm-up routines

20 A muffin recipe calls for ~ tablespoon of vanilla extract for 6 5

muffins Arthur is making 18 muffins How much vanilla extract does he need

Chapter 3 133 Practice and Problem Solving

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 Ariels English homework is to read

24 pages She reads 81 of the

assignment on the bus ride home How many pages does she read on the bus

3 As of 1990 the American Indian population was about 2000000

1About - were Cherokee About how

5 many members of the Cherokee tribe were there in 1990

Choose the letter for the best answer

5 One cup of dry dog food weighs 4 1

1- ounces A K9 dog eats 6 - cups5 3

of food a day How many ounces of food does the dog eat each day

A 8

4 ounces C 2

11 ounces 15 5

B 2

8- ounces 15

14 o 3- ounces27

7 A meteorite found in Willamette

Oregon weighed 1~ as much as one

found in Armanti Western Mongolia The meteorite found in Armanti weighed 22 tons How much did the one in Oregon weigh

1 A 31 5 tons C 22- tons

10

B 212 tons o 15 2 tons 10 5

Chapter 3 134

2 When a group of 40 friends goes to

the movies at a multiplex 1 of the 5

group decides to watch a science fiction movie How many of the group see the science fiction movie

4 Ron spends 3 hours painting a

picture Ashley spends 23 as long 3

creating a sculpture How long does Ashley work on her sculpture

6 Max has enough chicken to make

8 ~ servings of salad He needs i pound of chicken per serving How much chicken does he have

5F 8 pounds H 8 8 pounds

1 1 G 116 pounds J 2 2 pounds

8 Nicole takes part in a 12 -mile walkshy2

a-thon to raise money for charity She 1

stops 2 of the way to rest How much

farther must Nicole walk to finish the walk-a-thon

F 6 miles H 12 miles 4

G 6 miles J 25 miles 2

Practice and Problem Solving

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name__________________ _______ Class

Dividing Fractions and Mixed Numbers

Essential question How do you divide fractions and mixed numbers

Two numbers are reciprocals or multiplicative inverses if their product is 1 Dividing by a fraction is the same as multiplying by its reciprocal

~ I

_EXPLORE Dividing Fractions

One serving ofuncooked rice is ~ cup Noah pours out i cup of uncooked rice from a box What is the change in the number of servings of rice in the box

A Set up a division problem

You need to find how many groups of-- cup are in ~ cup

Use a negative number to represent the number of cups removed

Find -~--

B Rewrite the division problem as a multiplication problem

Multiply by the reciprocal of~

C Rewrite the division problem as a multiplication problem

The first factor is negative and the second factor is positive so the product will be negative positive

o Multiply Simplify the product and write it as a mixed number

_l~= ___ = ___ = 4 ]

The amount of rice in the box decreases increases by ___ servings

TRY THiS

Find each quotient

1a (-3) = 1b 4 + l~ =~____

REfLECT

1d Explain how could check your answer to 0

Chapter 3 135 Lesson 7

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Quotients can have negative signs in different places

Quotients and Placement of Negative Signs

Are the rational numbers ~~ -~2 and -(1f) equivalent

Amiddotmiddot Find each quotient Then use the rules for integer division to make sure the sign of the quotient is correct

-12~= -4-=--- -(If) = _____-4 -- shy

B What do you notice about each quotient

C The rational numbers are are not equivalent

1) Conjecture Explain how the placement of the negative sign in the rational number affects the sign of the quotients

Write equivalent expressions for each quotient

-32 -992b 2c-8 ~-- -9----~--

Quotients of Rational Numbers

Find each quotient 3

A 8-1 -4 The quotient will be negative positive because the signs are ___________

To find the quotient rewrite as ~ x _____________________

x8

=

Chapter 3 136 Lesson 7

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Find each quotient

B

The quotient will be negative positive because the signs are __________

REFLECT 1

3 Explain why the expression ~ is equivalent to ~ +- i 8

~a-EXAMP L E Fractions and Average Speed

It takes Anna l minute to ride her bike a distance of ii mile What is Annas average speed in miles per minute

distance in miles Think Annas average speed is

divided by

time in minutes

To find the quotient rewrite the expression as kX __

lx-_= __=-shy8

Annas average speed is mile per minute

REFLECT

4a Explain how you know that your answer is reasonable

4b What if What if it took Anna t minute to ride her bike kmile What would be her average speed in miles per minute

Chapter 3 137 Lesson 7

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

PRACTICE

Find each quotient

1 + (-6) = 2 3 + (-~) 3 Ii + 2i= 2

-81 -4

3

4 4 5 5 -2= 6 -1

5 3 -6

Write two equivalent expressions for each quotient by changing the placement of negative signs

-1 77 8 9 3--- ---- -12--- -- shy

Write and simplify an expression to solve each problem

10 Joy has ~ yard of fabric She cuts the fabric into pieces that are each ~ yard long How many pieces of fabric does Joy have

11 A hikers elevation changed by an average of - 2~ feet per minute as he went down a hill His total change in elevation was -35 feet How many minutes did it take the hiker to go down the hill

12 An insect crawls ~ foot in i minute What is the insects average speed in feet per minute

13 When you divide a positive mixed number by a positive fraction less than 1 is the quotient less than or greater than I Explain

5 -G 5 1

What error did Eric make What is the correct quotient

14 Error Analysis Eric simplified the expression as shown

- = -- Xshy1 3 5

- -18

Chapter 3 138 Lesson 7

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ____________ Class _____ Date____illl

_ Additional Practice) Divide Write each answer in simplest form

1 1 1 1 314+ - 2 - +- 3 - +shy

2 5 4 3 5

8 2 3 3 7 34 - +- 5 -- + - 6 - +shy

9 3 8 4 10 5

5 2 3 4 7 37 - +- 8 - +- 9 - +shy

12 5 4 9 12 4

1 1 1 2 1 110 -4- + - 113-+- 126- + shy

6 3 4 5 9 6

1 3 3 5 1 4132-+1- 143- + 2- 155- +-1shy

4 4 4 6 3 5

1 1 3 1 2 1162-+2- 17 -1- + 1- 187-+1shy

2 3 4 4 3 5

219 Burger Barn has 46- pounds of ground beef How many

3

~ -pound burgers can be made using all the ground beef 3

20 Roberto needs some roofing tiles to be cut from a large tile

How many tiles that are each 14 ~ inches in length can he cut 8

from a larger piece of tile that is 1 00 ~ inches long 8

Chapter 3 139 Practice and Problem Solving

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 The Wheeling Bridge in West Virginia

is about 307 i meters long If you 5

walk with a stride of about ~ meter 10

how many steps would it take you to cross this suspension bridge

3 A hiking trail is 6 2 miles long It has 3

4 exercise stations spaced evenly along the trail What is the distance between each exercise station

Choose the letter for the best answer

5 Matt wants to decorate his skateboard with decals His

skateboard is 28 3 inches long The 4

1 decals are 5 - inches long If Matt

2 arranges them in a line from the front to the back how many decals will fit

A 5 decals C 7 decals

B 6 decals D 8 decals

7 Bev buys a sleeve of ball bearings for her skateboard Each of the

bearings is 1 inches wide The 5

sleeve is 9~ inches long How many5

bearings are in the sleeve

A 5 bearings C 9 bearings

Chapter 3 140

2 The Flathead Rail Tunnel in Montana

is about 7 ~ rniles long If a train 4

travels through the tunnel at a speed

of about 1 miles per minute how 2

long will it take to pass from one end of the tunnel to the other

4 Jamal buys a strip of 25 postage stamps The strip of stamps is

21 7 inches long How long is each 8

stamp

6 A square floor tile measures

3 square feet How many tiles are 4 required to cover a 200 square foot floor

F 150tiles H 275 tiles

G 267 tiles J 300 tiles

8 The average hamster weighs about

pound The total weight of all the 4 hamsters in a cage in a pet store is

11 - pounds How many hamsters are

2 in the cage

F 3 hamsters H 5 hamsters

G 4 hamsters J 6 hamsters

Practice and Problem Solving

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

gtshyc 0shy E 0

U 0gtc

~ 0 gt shy~

g gt

t c

1E ~ c 0 +c 0gt gt 0 t copy

Name______________ Class_______

Solving Equations Containing Fractions

Essential question How do you solve equations containingfractions

Addition and Subtradhm Equations

A Jeff s puppy has gained ~ pound since he got it It now weighs ~ pounds

The equation w +~ =6i can be used to find w the puppys weight when

Jeff got it How much did the puppy weigh when Jeff got it

w+l=g4 2

w + ~ = 6-- Write all fractions with a common denominator

Subtraction Property ofEquality

w = Solve for w

Jeffs puppy weighed ___ pounds when he got it

B In June a city normally receives i inch of rain This June the city received i inch below

normal The equation a - i = - i can be used to find a the amount of rain the city

received this June How many inches of rain did the city receive this June

a - i = - -- Write all fractions with a common denominator

-- + -- Addition Property ofEquality

a = -- Solve for w

The city received _~___ inch of rain this June

TRY

Solve each equation 2 5 n= _____ x= ____1a 3 + n = -6 1b

Chapter 3 141 Lesson 8

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

1c Estimation How can you use estimation to show that your answer to problem Ib is correct

You can use the ~M~ii~l~~ll~l~~~Jf~~j~Jto solve multiplication equations that contain fractions

The product of a nonzero number and its multiplicative inverse or reciprocal is 1

Solving Multiplication and Division Equations

bull1 Kendra uses It cups of cornmeal to make each batch of cornbread

She uses a total ofsi cups of cornmeal The equation Itb = si can

be used to find b the number of batches Kendra made How many

batches of cornbread did Kendra make

Write improper fractions as mixed numbers

Multiplication Property ofEquality

Multiplicative Inverse Property ~ =___

Solvefor b

Kendra made ___batches of cornbread

B Armando cut a length of rope into 6 equal pieces The length of

each piece is j yard The equation ~ = j can be used to find X

the length of the original rope How long was the original rope

~ Multiplication Property ofEquality

Chapter 3 142 Lesson 8

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

c ~ J

~

x=----= Solveforx

The original rope was _____ yards long

a E X ~ M P l E Solving Problems Using Equations

Urszula is preparing for a long-distance race Today she ran 4i miles This distance is 1 times the number of miles she ran a week ago How many miles did Urszula run last week

A Write an equation to represent the dis

sent the unknown distance

tance Urszula ran last week Use a variable to represhy

B

Distance today

equals

Change the mixed numbers to impro

times

per fractions in the

Distance last week

equation

C Solve the equation

--=x

--=X

Urszula ran

REFLECT

Multiplication Property ofEquality

Solvefor x

Change the improper fraction to a mixed number

miles last week

2 To solve the equation ~n = ~ what number should you multiply both sides of the equation by Explain how you know which number to use

Chapter 3 143 Lesson 8

01

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

---- ---- ----

---- ---- ----

PRACTICE

Solve each equation

1 22 - C+ 715- lO 2bull g-34 = 13 3 x+ l~ -~

4 ~t = ~ 5 ~ =-~ 6 -~P=-2

The table shows the ingredient list for making a loaf of peanut butter bread Use the list to solve each problem

7 Marcia has 2~ cups of flour Write and solve an equation to fmd a the additional amount of flour she needs to make a loaf of peanut butter bread

3~ cups flour

11 cups water

2 tablespoons honey

1~ teaspoons yeast

~ cup peanut butter ~ teaspoon salt

8 Charlie used a total of 3 teaspoons of salt when making peanut butter bread Write and solve an equation to find b the number of loaves he made

9 Ruth used only 11 tablespoons of honey to make a loaf of peanut butter bread Write and solve an equation to find f the fraction of the full amount of honey that Ruth used

10 Explain how you know that your answer to problem 9 is reasonable

11 Give an example of a real-world situation that could be represented by the equation x i ~ Then solve the equation and tell what the solution means in the context of the situation

Chapter 3 144 Lesson 8

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ____________ Class_____ Date ____

Additio-Solve Write each answer in simplest form

3 4 5 3 3 21 t - - = - 2 9 - - = - 3 k- - =shy

7 7 16 16 10 5

1 2 5 34 n + - =- 5 j+ ~ = 17 6 f+ - =shy

7 3 6 18 12 4

1 3 1 17 -S= - 8 -a =- 9 i h = ~

4 4 5 2 5 9

2 5 3 7 2 1 10 P - - = - 11d--=- 12 y- - = 3shy

3 8 5 10 7 4

5 1 4 1 6 313 c+ - =2- 14w+-=3- 15z+-=2shy

12 6 15 3 7 5

5 8 1 7 1 216 -m=- 17 -x=3- 18 -r=2shy

6 9 2 15 5 3

19 Sarabeth ran 1 ~ miles on a path around the park This was ~ 5 8

of the distance around the park What is the distance around the park

20 An interior decorator bought 12 ~ yards of material to make

drapes He used 8 ~ yards on 1 pair of drapes How much 3

material does he have left

Chapter 3 145 Practice and Problem Solving

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Write the correct answer

1 At the 2002 Winter Olympics Austria

won 2 gold medals This was 1 of 8

the total medals Austria won How many medals did Austria win

3 Jesse plays ice hockey He is on the

ice ~ of each game A game lasts 5

45 minutes divided into 3 periods How many minutes is Jesse on the ice during each game

Choose the letter for the best answer

5 At the 2002 Winter Olympics China won 8 medals The United States

won 4 times as many medals as 4

China did How many medals did the United States win

A 24 medals C 34 medals

B 32 medals 0 40 medals

7 Yoriko ran 6 1 laps of the track Each 2

lap is of a mile How many miles 4

did she run

A 6~ miles C 2 miles

B 4 ~ miles o 1~ miles 8

Chapter 3

2 At the 2002 Winter Olympics Germany won 35 medals of which 16 were

silver They won 1 i times as many

silver medals as gold medals How many gold medals did Germany win

4 Amelias soccer team won ~ of its 4

games The team won 18 games How many games did the team play

6 Juanitas water bottle contains

1~ liters She uses her bottle to fill

Felixs When she is done her bottle

contains liter How many liters can 4

Felixs bottle hold

F ~ liter H 1~ liters

G 1 liters J 2i liters8

8 Rocky runs 3 miles each week 2

Leroy runs 5 i miles each week How

much farther does Leroy run

F 1 ~ miles H 2 ~ miles

5G 1 miles J 2 miles

6 4

146 Practice and Problem Solving

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name __________ Class______ Date___

Problem Solving Connections Camp Fun Waterfall Summer Camp has opened up for the season Several boys and girls are going this week You will use addition subtraction multiplication and division of rational numbers to learn about life at camp

D Cooking The camp chef is making vegetarian chili for the campers dinner The recipe he has serves 6 people He needs to make this dish for the campers and counselors-a total of 66 people Help him change the recipe to amounts that will make enough chili for 66 people

Vegetarian Chili (6 servings)

3i cups stewed tomatoes 1t teaspoons ground cumin

1i cups salsa ~ teaspoon salt

2 cups kidney beans

A Find the new amounts

B Rewrite the recipe for 66 servings

(~~~ON CORE

CONSJ CONS2 COIllS3

Vegetarian Chili (66 servings)

cups stewed tomatoes teaspoons ground cumin

cups salsa teaspoons salt

cups kidney beans

Chapter 3 147 Problem Solving Connections

I

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

o Weather Waterfall Summer Camp is located in the mountains and is open from April to October The table shows average monthly temperatures at the campsite

Apr May Jun Jul Aug Sept

25 70 65 85 92 82

Oct

A What is the average monthly temperature for the period of April through September Round your answer to the nearest tenth Show your work

B The local newspaper reported that the average monthly temperature for April through September was not typical of the temperatures at the campsite What do you think is the reason for this How could you find a more typical representation of the average monthly temperature for the campsite

C Find a more typical average for the monthly temperature

D The average monthly temperature from April to October was 70 of Find the average temperature for October and write it in the table

Chapter ~ 148 PrQblemSolvingmiddotConneruons

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

--------------

----------------------------------------

a Golf Some of the campers play 9 holes of golf one day Par or the expected number of strokes for the 9-hole course is 32 Every hole has a par value A golfers score is the number of strokes above or below par A golfers score is above par if he or she takes more strokes than the par value of the hole A golfers score is below par if he or she takes fewer strokes than the par value of the hole A score below par is indicated by a negative number

Round of 9-Hole Golf

bull 1)f

Golfer Jan Sergei Sue Yanni Raz Leon Felicity

Score -6 6 3 -4 2 -3 -5

A Which golfer had the best score Explain Now order the players from best score to worst score

B Which of the golfers have scores that are additive inverses Explain

C What is the average score of all the golfers

o Troys score is 2 times Leons score What is Troys score

E Twice the sum of 4 and Yannis score is Counselor Trinas score What is Counselor Trinas score

Chapter 3 149 Problem Solving Connections

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

D Hiking The campers go for a hike using the map shown below

Playground

A The distance from the cabins to the horse farm is one-third of the distance from the lake to the playground How would you find the distance from the cabins to the horse farm

8 How far is it from the cabins to the horse farm

C Junie walks from the cabins to the lake and back She then walks to the playground and back She does each trip twice a week How much does she walk in total in one week

150

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name __________ Class _______ Date ___

Performance Task

1 A website uses stars to rate movies The ratings are to the nearest quarter of a star A drama has a rating that is Ii stars more than a western The drama has a rating of 3~ stars

a Write and solve an equation to find the rating of the western

=-- CO~MON CORE

CC7NSJ CC7NS2 CC7NSJ CCZEE4

b Fran will not rent a movie if it has fewer than two thirds as many stars as her favorite movie Her favorite movie has a rating of 3~ stars Would she rent the western Explain

it l3 E 3 2 Srinias is marking off 35 inch portions of a loaf of bread He finds the ~ bread will be split into exactly 5 equal pieces Write and solve an g equation to find the length of the loaf 0shy

t

sect f

J

5 E

~ c 2c g o J

Chapter 3 151 Performance Task

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

3 Candace has $155 in quarters dimes and nickels She has the same bull number of quarters as dimes The dimes are worth $030 in all

a How many coins are dimes or quarters Explain

h What is the total value of the nickels Justify your answer

c Write and solve an equation to determine how many nickels Candace has

bull 4 Gwen has a rectangular piece of paper She folds it in half twice once widthwise and once lengthwise The folded paper is 5 inches long by 8 inches wide bull a How much longer is the length of the paper than the width before it

was folded Explain

h What would the dimensions of the folded paper have been ifGwen had folded lengthwise twice Show your work

152 Peffofl11an~(task

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

Name ____________ Class ______ Date ____

SELECTED RESPONSE

1 Misha has a board that is 17 ~ inches long 6 Sean bought 48 pounds of beef His total that he has to cut into 3 equal pieces How was $1176 before tax How much did Sean long should each piece be pay per pound of beef

A 5~inches F $245 H $1656

B 5~inches G $696 J $5645

C 8~ inches 7 At the end of the year Juan has $5271 D 14~ inches more than 4 times his balance at the

beginning of the year If his ending 2 Jenni buys a piece of fabric that is 4k yards balance is $17290 what was his balance

long What is the decimal equivalent of at the beginning of the year 4 71

8 A $3005 F 0875 H 4875

B $5641 G48 J495

C $9594

3 A girl is snorkeling 1 meter below sea level D $12020 and then dives down another 05 meter How far below sea level is the girl 8 Tim and Todd are twins At birth Todd

weighed 1 t pounds more than TimsA 05 meter below sea level weight IfTodd weighed 7~ pounds at

B 1 meter below sea level birth how much did Tim weigh at birth

C 15 meters below sea level c gt F 41pounds Q E D 2 meters below sea level G 6~pounds0 v

H7tPOundsc 4 A mountain climber climbed up a cliff ~

J3 i mile at a time He did this 5 times in one J 8~poundsl 0shy

1 l day How many miles did he climb up 0

9 Jody is cutting wood to make picturer F tmile s frames How many pieces of wood that are ~ G ~mile 10~ inches long can she cut from a board c that is 75~ inches long0 1 H 1 mile l A 7 pieces r0 J It miles

B 10 pieces 5 Umas salad dressing recipe calls for ~ cup

C 65 pieces of yogurt She wants to triple her recipe How much yogurt does she need D 85 pieces

A l~ cups C 3 cups

B 2 cups D 3~ cups

Chapter 3 153 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness

10 Jim raised $4364 for the community food bank Rina raised $16523 for the food bank and Everett raised $2309 for the food bank How much did they raise together

F $6673

G $16523

H $20887

J $23196

11 A new poll shows that 191 of all students like to eat pizza Jan wants to write I as a decimaL What is I in decimal form

A08

B081

C 08181

D 090

CONSTRUCTED RESPONSE

12 Sheila has to pack 128 baskets of apples

She has packed tof the baskets How

many baskets are left for her to pack

13 Leanne was asked on a math test if the number 058 is a rational number She says it is not a rational number Is she correct Why or why not

14 Is the rectangular room shown below a square Explain your answer

15 Mr Carrara has a r~d car and a blue car He measured the volume of each cars gas tank and each cars mileage Find the distance each car can travel on a single tank of gas Which car travels farther on a single tank Explain

Red 19 gal 253 migal

Blue 16 gal 284 migal

16 A grocery store advertises cherries on sale for $2 per pound At a different store Millie paid $11 for 2 pounds of cherries How much more did Millie pay per pound for cherries than if she had bought them on sale Explain

Chapter 3 154 Assessment Readiness